Stems cells from the umbilical cord are refered to as embryonic stem cells. These cells are totipotent which refers to their unlimited ability to differentiate into any type of cell, as long as they recieve the correct signals and all the various factors are in place.

Adult stem cells are pluripotent which basically means that they can differentiate into many different cells but are very limited in their ability to do so. Just think of the stem cells in your bone marrow, they can differentiate into various types of blood cells but can not ever differentiate into a nervous tissue cell.